[fitswcs] I've got this FITS WCS matrix

[Mark Calabretta]

On Sat 2004/08/21 03:15:23 +0100, "Malcolm J. Currie" wrote
in a message to: fitswcs at aoc.nrao.edu

Malcolm,

>> Suppose that a CDi_j matrix is the product of a diagonal (scaling)
>> matrix times a rotation matrix.
>
>Like CDELTi and the PC matrix.

The PC matrix need not be a rotation.

>Thanks for spelling it out.  It could be useful when I contact the
>Gemini people about it.

Previously I argued from the sign information alone that this matrix
cannot be factorized as a scale * rotation.  That provided a handy
heuristic that allowed an immediate conclusion to be drawn without
recourse to a calculator.

However, it's *not* just the signs that preclude this factorization;
there's no way of changing signs or permuting matrix elements that can
turn this into a scale * rotation.

Here is the factorization that makes the diagonal elements unity:

( 6.027e-06  1.280e-08) = (6.027e-06  0.0      ) (1.0       0.0021230)
(-1.606e-08 -5.968e-06) = (0.0       -5.968e-06) (0.0026905 1.0      )

Notice that the off-diagonal elements are not equal in absolute value
as they would have to be for a rotation matrix.  That is not to say that
this matrix is wrong (it looks plausible to me) only that you cannot
factorize it as you would like.

It's only because the off-diagonal terms are small compared to the
diagonal terms that you can get a factorization as a scale * rotation
that is not obviously wrong.

The question of how any given matrix would best be approximated as a
scale * rotation is a separate and more complicated issue.  (In this
instance the best rotation angle is very close to zero.)

Cheers, Mark